Postprocessing the Galerkin method:: The finite-element case

A postprocessing technique, developed earlier for spectral methods, is extended here to Galerkin finite-element methods for dissipative evolution partial differential equations. The postprocessing amounts to solving a linear elliptic problem on a finer grid (or higher-order space) once the time integration on the coarser mesh is completed. This technique increases the convergence rate of the finite-element method to which it is applied, and this is done at almost no additional computational cost. The numerical experiments presented here show that the resulting postprocessed method is computationally more efficient than the method to which it is applied (say, quadratic finite elements) as well as standard methods of similar order of convergence as the postprocessed one (say, cubic finite elements). The error analysis of the new method is performed in L-2 and in L-infinity norms.